Dzeko's effort meant Roma were through to the quarter-finals on away goals having lost the first leg in Ukraine 2-1.
Shakhtar's European hopes suffered another blow 11 minutes from time with the dismissal of Ordets for hauling down Dzeko on the edge of the box as he broke through on goal, although Kolarov let them off after firing his free-kick straight into the wall.
The Bosnian took a defence-splitting pass from midfielder Kevin Strootman into his stride and beat the offside trap before he steered the ball into the far corner with the outside of his foot past advancing keeper Andriy Pyatov.
And in the Roma vs. Shakhtar Donetsk game in Rome, William Hill's bet for the Italian team winning this encounter is 1.67.
The two teams went into the interval level, with the visitors looking the more unsafe of the two in the opening stages. Substitute Wissam Ben Yedder scored twice after coming on in the second half, with Romelu Lukaku pulling a goal back for United.
Alessandro Florenzi flicked a risky backwards header just wide of his own goal and a nervous home crowd witnessed the visitors maintaining a level of control throughout the first half, Pyatov intercepting when Aleksandar Kolarov led a rare Roma foray.
Ordets was shown a red card for tugging on Dzeko's shirt before a scuffle broke out on the pitch after two Shakhtar players pushed a ballboy over the advertising boards for holding on to the ball. "We managed to do something Roma hadn't in 10 years, so going forward to play against the best makes us all very proud".
'We've won five of our last six in the league and we're into the last eight in Europe, so I am seeing growth in this team, ' the Roma coach said, as quoted on www.uefa.com.
Dzeko was given a standing ovation when he was substituted in the final minute. "Everyone gave their all and we are deservedly in the quarterfinals", Dzeko said.